Critical Point for Bifurcation Cascades and Featureless Turbulence

In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyze the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures (pipe-to-torus diameter ratio) and supercritical transition at high curvatures, as was previously documented. We unveil an additional step in the bifurcation cascade and provide evidence that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emerging through subcritical transition mechanisms.

Figure 1

Portion of the $\delta$-Re parameter space of the flow in a toroidal pipe. Experimental and numerical data from the literature are reported as well as the location of the present computations. The gray shaded area indicates the approximate boundaries of the critical point, where subcritical transition and a bifurcation cascade coexist. Filled downward-pointing triangles indicate spatially expanding turbulence, while empty triangles relaminarization. The data from Sreenivasan and Strykowsky (Ref. [37]) are the curve they refer to as the “conservative lower critical limit”. The data from Kühnen et al. (Ref. [35]) denote 50% intermittency at low curvatures and the appearance of the supercritical traveling wave above the critical point.

Figure 2

Power spectral densities (PSDs) and corresponding phase space representations for point velocity measurements at $\delta =0.05$ and $\mathrm{Re}=4000$ (a), $\mathrm{Re}=4500$ (b), and $\mathrm{Re}=6000$ (c). Yellow lines represent the PSDs of radial velocity $u_r$, while blue lines that of the azimuthal component $u_{\theta }$.

Figure 3

Snapshots of the flow for $\delta =0.022$, $\mathrm{Re}=5050$ along the three trajectories in Fig. 4. (a) corresponds to the (blue) traveling wave, (b) to the intermediate (orange) trajectory that returns to the traveling wave, while (c) is along the (red) trajectory that converges to sustained turbulence.

Figure 4

Trajectories in the kinetic energy-friction factor phase space at the critical point, $\delta =0.22$, $\mathrm{Re}=5050$. Arrowheads indicate the direction of time, and $E_k$ and $f$ are normalized by their respective values for the (unstable) laminar flow. The filled circle, plus, and cross markers indicate the location of the snapshots in Fig. 3. It is expected that for turbulent flow both $E_k$ and $f$ are lower than for the laminar flow, as this curvature range is subject to sublaminar drag [48]. Being just above the neutral curve, $\mathrm{Re}=5013$ at this curvature, the traveling wave is still very close ($<1\%$) to the unstable laminar flow. Letter (a)–(c) are explained in the text.

Figure 5

Sketch of the phase space at criticality. The left pane illustrates the behavior of the system before the bifurcation: the steady state is stable and turbulence is reached via subcritical transition—point $A$ in Fig. 1. The right pane shows the state of the system after the supercritical Hopf bifurcation: The steady state is now unstable and the first step of a bifurcation cascade has appeared in the form of a traveling wave—point $C$ in Fig. 1, illustrated in detail in Figs. 3 and 4.